D-branes on Calabi-Yau manifolds and helices

TL;DR

The paper addresses translating D-brane data between large-volume Calabi–Yau geometry and orbifold points using the McKay correspondence and Beilinson’s derived equivalence. It introduces two conjectures: a computational improvement to build McKay-type dictionaries via tautological bundles and helices, and a framework linking helices and mutations to Gram–Schmidt orthogonalization. The authors verify these ideas in explicit toric Calabi–Yau examples with multiple fibration structures, constructing exceptional collections R_i and their mutations to realize a Beilinson-type dictionary and to compare with McKay quivers. The work clarifies how derived-category methods organize brane charges, provides a physically motivated derivation of why crepant toric resolutions align with quiver descriptions, and connects these structures to mirror symmetry through the helices perspective.

Abstract

We investigate further on the correspondence between branes on a Calabi-Yau in the large volume limit and in the orbifold limit. We conjecture a new procedure which improves computationally the McKay correspondence and prove it in a non trivial example. We point out the relevance of helices and try to draw some general conclusions about Beilinson theorem and McKay correspondence.